Special Talks Perry County Schools
Perry County schools offer specialized services for children with disabilities, including autism, developmental delay, deaf/blindness, emotional disability, and more. The dedicated team of professionals provides training, support, and resources to help parents and students navigate the special education process. Find guidance on dealing with different disabilities and practical tips for supporting children with unique needs.
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01 March 2025 Circle theorems: Angles at the centre and at the circumference on the same arc LO: To solve problems using circle theorems. www.mathssupport.org
Naming the parts of a circle A circle is a set of points equidistant from a fixed point called the centre. The distance around the entire circle boundary is called the circumference. The radius is any line segment joining its centre to any point on the circumference. radius centre The diameter is a line segment passing through the centre. Note that the diameter of a circle is twice its radius circumference www.mathssupport.org
Naming the parts of a circle A chord is any line segment that joins two points on the circle. Therefore, a diameter is an example of a chord. It is the longest possible chord. The line that touches the circumference in exactly one point is called a tangent or a tangent line. The point where the tangent touches the circle is called the point of contact or point of tangency. centre www.mathssupport.org
Naming the parts of a circle The region of a circle enclosed by a chord and an arc a Segment is called Any chord encloses two segments, which have different areas. If the segment is enclosed by the diameter a semicircle Major segment it is called Minor segment www.mathssupport.org
Arcs and sectors arc An arc is any continuous part of the circumference. sector When an arc is bounded by two radii a sector is formed. www.mathssupport.org
The angle at the centre Draw a circle, label the centre O Choose any three points on the circumference and label them A, B and C B Draw a line from A, to the centre O, Draw a line from C, to the centre O, x Measure the angle AOC, O Draw a line from A, to the point B, Draw a line from C, to the point B, 2x C Measure the angle ABC, A What do you notice? So, angle AOC = 2 x angle ABC Statement The angle at the centre is twice the angle at the circumference www.mathssupport.org
The angle at the centre We have just seen a demonstration that shows that the angle at the centre of a circle is twice the angle at the circumference. We can prove this result as follows: Draw a line from B, through the centre O, and to the other side D. B x In triangle AOB, (both radii) OA = OB O So, angle OAB = angle OBA x C (angles at the base of an isosceles triangle) A D Let s call these angles x. www.mathssupport.org
The angle at the centre We have just seen a demonstration that shows that the angle at the centre of a circle is twice the angle at the circumference. We can prove this result as follows: In triangle BOC, B (both radii) OB = OC y x So, angle OBC = angle OCB O y (angles at the base of an isosceles triangle) x C Let s call these angles y. A D www.mathssupport.org
The angle at the centre We have just seen a demonstration that shows that the angle at the centre of a circle is twice the angle at the circumference. We can prove this result as follows: angle AOD = 2x angle COD = 2y B y x (the exterior angle in a triangle is equal to the sum of the opposite interior angles) angle AOC = 2x + 2y = 2(x + y) angle ABC = x + y angle AOC = 2 angle ABC O y 2x2y x C A D www.mathssupport.org
Calculating the size of unknown angles Calculate the size of the labelled angles in the following diagram: a = 29 (angles at the base of an isosceles triangle) c b = 180 2 29 = 122 (angles in a triangle) O c = 122 2 = 61 (angle at the centre is twice angle on the circumference) b 41 a d 29 d = 180 (29 + 29 + 41 + 61 ) = 20 (angles in a triangle) www.mathssupport.org
Angles subtended by the same arc Draw a circle, label the centre O. Choose any four points on the circumference and label them A, B, C, D: Draw a line from A to D. C Draw a line from B to D. D Measure the angle ADB Draw a line from A to C. O Draw a line from B to C. B Measure the angle ACB Compare the angles ADB and ACB A What do you notice? angle ADB = angle ACB Statement Angles subtended by an arc on the circle on the circle are equal in size. www.mathssupport.org
Angles subtended by the same arc We have just seen a demonstration that shows that the angles in the same segment are equal. We can prove this result as follows: C Mark the centre of the circle O and show angle AOB. D angle ADB = of angle AOB O and angle ACB = of angle AOB B (the angle at the centre of a circle is twice the angle at the circumference) A angle ADB = angle ACB www.mathssupport.org
Calculating the size of unknown angles Calculate the size of the labelled angles in the following diagram: a = 90 51 = 39 (angle in a semi-circle) c b = 180 (90 + 44 ) = 46 (angles in a triangle) d O 44 c = b = 46 (angles in the same segment) b a 51 d = 51 (angles in the same segment) www.mathssupport.org
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